Lie Symmetrical Non-Noether Conserved Quantities of Poincarè-Chetaev Equations

In the present paper the Lie symmetrical non-Noether conserved quantity of the Poincarè-Chetaev equations under the general infinitesimal transformations of Lie groups is discussed. First, we establish the determining equations of Lie symmetry of the equations. Second, the Lie symmetrical non-Noether conserved quantity of the equations is deduced.     

Lie Symmetrical Non-Noether Conserved Quantity of Mechanical System in Phase Space

In this paper, we study the Lie symmetrical non-Noether conserved quantity of the differential equations of motion of mechanical system in phase space under the general infinitesimal transformations of groups. Firstly. we give the determining equations of the Lie symmetry of the system. Secondly, the non-Noether conserved quantity of the Lie symmetry is derived. Finally, an example is given to illustrate the application of the result.     

Non-Noether Conserved Quantity of Poincaré-Chetaev Equations of a Generalized Classical Mechanics

In the present paper the Lie symmetrical non-Noether conserved quantity of the Poincaré-Chetaev equations of a generalized classical mechanics under the general infinitesimal transformations of Lie groups is discussed. First, we establish the determining equations of Lie symmetry of the equations. Second, the Lie symmetrical non-Noether conserved quantity of the equations is deduced. Finally, an example is given to illustrate the application of the results.     

A new non-Noether conserved quantity of the relativistic holonomic nonconservative systems in general Lie transformations

For the relativistic holonomic nonconservative system, a new Lie symmetrical non-Noether conserved quantity is given under general infinitesimal transformations of groups in which time is variable. On the basis of the theory of invariance of differential equations of motion under infinitesimal transformations for t and qs, we construct the relativistic Lie symmetrical determining equations and obtain directly a new relativistic Lie symmetrical non-Noether conserved quantity of the system, which only depend on the variables t, qs and qs. An example is given to illustrate the application of the results.     

变质量Birkhoff系统的Lie对称性和非Noether守恒量

采用嵌入质量法建立了变质量系统的Birkhoff方程.根据Lie对称性理论给出了变质量Birkhoff系统的Lie对称性确定方程，得到了系统的Lie对称直接导致非Noether守恒量的存在条件和形成.举例说明结果的应用. 关键词： 变质量 Birkhoff系统 Lie对称性 非Noether守恒量     

A new type of Lie symmetrical non-Noether conserved quantity for nonholonomic systems

For a nonholonomic system, a new type of Lie symmetrical non-Noether conserved quantity is given under general infinitesimal transformations of groups in which time is variable. On the basis of the invariance theory of differential equations of motion under infinitesimal transformations for t and q_s, we construct the Lie symmetrical determining equations, the constrained restriction equations and the additional restriction equations of the system. And a new type of Lie symmetrical non-Noether conserved quantity is directly obtained from the Lie symmetry of the system, which only depends on the variables t, q_s and \dot{q}_s. A series of deductions are inferred for a holonomic nonconservative system, Lagrangian system and other dynamical systems in the case of vanishing of time variation. An example is given to illustrate the application of the results.     

A set of Lie symmetrical non－Noether conserved quantity for the relativistic Hamiltonian systems

For the relativistic Hamiltonian system, a new type of Lie symmetrical non-Noether conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing special infinitesimal transformations for q_s and p_s, we construct the determining equations of Lie symmetrical transformations of the system, which only depend on the canonical variables. A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.     

Non-Noether Conserved Quantity of Nonholonomic System Having Variable Mass and Unilateral Constraints

Using the Lie Symmetry under infinitesimal transformations in which the time is not variable, the nonNoether conserved quantity of nonholonomic system having variable mass and unilateral constraints is studied. The differential equations of motion of the system are given. The determining equations of Lie symmetrical transformations of the system under infinitesimal transformations are constructed. The Hojman‘s conservation theorem of the system is established. Finally, we give an example to illustrate the application of the result.     

New non-Noether conserved quantities of mechanical system in phase space

This paper focuses on studying non-Noether conserved quantities of Lie symmetry and of form invariance for a mechanical system in phase space under the general infinitesimal transformation of groups. We obtain a new non-Noether conserved quantity of Lie symmetry of the system, and Hojman and Mei's results are of special cases of our conclusion. We find a condition under which the form invariance of the system will lead to a Lie symmetry, and, further, obtain a new non-Noether conserved quantity of form invariance of the system. An example is given finally to illustrate these results.     

Noether Symmetry Can Lead to Non-Noether Conserved Quantity of Holonomic Nonconservative Systems in General Lie Transformations

For the holonomic nonconservative system, by using the Noether symmetry, a non-Noether conserved quantity is obtained directly under general infinitesimal transformations of groups in which time is variable. At first, the Noether symmetry, Lie symmetry, and Noether conserved quantity are given. Secondly, the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations is obtained. Finally, a set of non-Noether conserved quantities of the system are given by the Noether symmetry, and an example is given to illustrate the application of the results.     

Non-Noether symmetries and conserved quantities of the Lagrange mechano-electrical systems

This paper focuses on studying non-Noether symmetries and conserved quantities of Lagrange mechano-electrical dynamical systems. Based on the relationships between the motion and Lagrangian, we present conservation laws on non-Noether symmetries for Lagrange mechano-electrical dynamical systems. A criterion is obtained on which non-Noether symmetry leads to Noether symmetry of the systems. The work also gives connections between the non-Noether symmetries and Lie point symmetries, and further obtains Lie invariants to form a complete set of non-Noether conserved quantity. Finally, an example is discussed to illustrate these results.     

变质量力学系统的一般形式的非Noether守恒量

研究一般的无限小变换下变质量力学系统Lie对称性的非Noether守恒量, 进一步推广Hojma n定理. 给出变质量力学系统的一般形式的非Noether守恒量,并举例说明结果的应用. 关键词： 变质量系统 一般的无限小变换 非Noether守恒量     

Noether symmetry and non-Noether conserved quantity of the relativistic holonomic nonconservative systems in general Lie transformations

For a relativistic holonomic nonconservative system, by using the Noether symmetry, a new non-Noether conserved quantity is given under general infinitesimal transformations of groups. On the basis of the theory of invariance of differential equations of motion under general infinitesimal transformations, we construct the relativistic Noether symmetry, Lie symmetry and the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations. By using the Noether symmetry, a new relativistic non-Noether conserved quantity is given which only depends on the variables $t$, $q_s $ and $\dot {q}_s $. An example is given to illustrate the application of the results.     

相对论性转动变质量非完整可控力学系统的非Noether守恒量

研究相对论性转动变质量非完整可控力学系统的非Noether守恒量——Hojman守恒量. 建立了系统的运动微分方程, 给出了系统在特殊无限小变换下的Mei对称性(形式不变性) 和Lie对称性的定义和判据, 以及系统的Mei对称性是Lie对称性的充分必要条件. 得到了系统Mei对称性导致非Noether守恒量的条件和具体形式. 举例说明结果的应用. 关键词： 相对论性转动 可控力学系统 变质量 非Noether守恒量     

Lie symmetries and non-Noether conserved quantities for Hamiltonian canonical equations

This paper focuses on studying Lie symmetries and non-Noether conserved quantities of Hamiltonian dynamical systems in phase space. Based on the infinitesimal transformations with respect to the generalized coordinates and generalized momenta, we obtain the determining equations and structure equation of the Lie symmetry for Hamiltonian dynamical systems. This work extends the research of non-Noether conserved quantity for Hamilton canonical equations, and leads directly to a new type of non-Noether conserved quantities of the systems. Finally, an example is given to illustrate these results.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantitiesare given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, andintroducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determiningequations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example isgiven to illustrate the application of the results.     

相对论性变质量非完整可控力学系统的非Noether守恒量

在时间不变的特殊无限小变换下,研究相对论性变质量非完整可控力学系统的非Noether守恒量——Hojamn守恒量.建立了系统的运动微分方程, 给出了系统在特殊无限小变换下的形式不变性(Mei对称性)的定义和判据以及系统的形式不变性是Lie对称性的充分必要条件.得到了系统形式不变性导致非Noether守恒量的条件和具体形式.举例说明结果的应用. 关键词： 相对论 非完整可控力学系统 变质量 非Noether守恒量     

Lie Symmetries and Non-Noether Conserved Quantities of Nonholonomic Systems

A new conservation theorem of the nonholonomic systems is studied. The conserved quantity is onlyconstructed in terms of a general Lie group of transformation vector of the dynamical equations. Firstly, we establish thedynamical equations of the nonholonomic systems and the determining equations of Lie symmetry. Next, the theore mof non-Noether conserved quantity is deduced. Finally, we give an example to illustrate the application of the result.     

Non-Noether Conserved Quantity for Relativistic Nonholonomic System with Variable Mass

Using form invariance under special infinitesimal transformations in which time is not variable, the non-Noether conserved quantity of the relativistic nonholonomic system with variable mass is studied. The differential equations of motion of the system are established. The definition and criterion of the form invariance of the system under infinitesimal transformations are studied. The necessary and sufficient condition under which the form invariance is a Lie symmetry is given. The condition under which the form invariance can be led to a non-Noether conserved quantity and the form of the conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determining equations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.